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Theorem symdifxor 3231
Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)
Assertion
Ref Expression
symdifxor  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Distinct variable groups:    x, A    x, B

Proof of Theorem symdifxor
StepHypRef Expression
1 eldif 2955 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 2955 . . . 4  |-  ( x  e.  ( B  \  A )  <->  ( x  e.  B  /\  -.  x  e.  A ) )
31, 2orbi12i 691 . . 3  |-  ( ( x  e.  ( A 
\  B )  \/  x  e.  ( B 
\  A ) )  <-> 
( ( x  e.  A  /\  -.  x  e.  B )  \/  (
x  e.  B  /\  -.  x  e.  A
) ) )
4 elun 3112 . . 3  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  ( A  \  B
)  \/  x  e.  ( B  \  A
) ) )
5 excxor 1285 . . . 4  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
) )
6 ancom 257 . . . . 5  |-  ( ( -.  x  e.  A  /\  x  e.  B
)  <->  ( x  e.  B  /\  -.  x  e.  A ) )
76orbi2i 689 . . . 4  |-  ( ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( -.  x  e.  A  /\  x  e.  B )
)  <->  ( ( x  e.  A  /\  -.  x  e.  B )  \/  ( x  e.  B  /\  -.  x  e.  A
) ) )
85, 7bitri 177 . . 3  |-  ( ( x  e.  A  \/_  x  e.  B )  <->  ( ( x  e.  A  /\  -.  x  e.  B
)  \/  ( x  e.  B  /\  -.  x  e.  A )
) )
93, 4, 83bitr4i 205 . 2  |-  ( x  e.  ( ( A 
\  B )  u.  ( B  \  A
) )  <->  ( x  e.  A  \/_  x  e.  B ) )
109abbi2i 2168 1  |-  ( ( A  \  B )  u.  ( B  \  A ) )  =  { x  |  ( x  e.  A  \/_  x  e.  B ) }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    \/ wo 639    = wceq 1259    \/_ wxo 1282    e. wcel 1409   {cab 2042    \ cdif 2942    u. cun 2943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-xor 1283  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950
This theorem is referenced by: (None)
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